Reconstruction of Grounded Objects from Cauchy Data on a Plane

Abstract

This thesis explores the use of capacitance measurements made between electrodes embedded in or around a display surface, to detect the position, orientation and shape
of hands and fingers. This is of interest for unobtrusive 3D gesture input for interactive displays, so called touch-less interaction. The hand is assumed to be grounded.
The forward problem is solved using Green’s theorem and an appropriate Green’s function. This leads to an operator factorisation for the forward Dirichlet to Neumann
map Lambda_D : L^2(boundary H) -> L^2(boundary H). The foward map is demonstrated to be compact, injective and depends uniquely on the object. An alternative factorisation based on double layer potentials and involving a Fredholm equation of the second kind is also presented. These operator expressions are used in numerical calculations in two and three space dimensions using the Boundary Element Method for discretization.
Four methods are presented for the solution of the inverse problem of recovering the object from a measured forward map. The first uses modified Gauss-Newton optimization.
The method is successful if the degrees of freedom are limited to object position, size and orientation, but is unpractical for shape reconstruction. The second method recovers the zero potential contour of a solution to Laplace’s equation from Cauchy data on part of the boundary of a domain. An algorithm is used where at each iteration there is an approximation boundary D_k to boundary D on which approximate Cauchy data are calculated by solving a Tikhonov regularised linear system. This
data is used to modify boundary D_k by extrapolation towards the zero-surface giving the next
approximation boundary D_{k+1}
In the third method the problem is solved with the so-called Factorisation Method.
A test function g_z is used to characterise points z in D iff g_z in Range(Lambda^{1/2}_D). Implicit
regularisation due to the finite aperture of the measurement electrode results in a level
set P(z) that is finite and differentiable everywhere. The level representing the object
boundary D is found through minimization of the cost function.
The fourth method uses a monotonicity property of the forward map to test if
a probe object is contained within the unknown object. For an infinitesimal probe
object and finite aperture measurements the method is shown to be identical to the
factorisation method.
The thesis closes with conclusions on the relative merits of these methods.